Nov 12, 2010

Optimal Trading Execution

You are trying to buy a stock at the best price. You need to buy it in the next 100 minutes. Every minute you will receive a random price (uniform distribution) that is a number between 1 and 100 dollars and decide whether to buy it or not.

1. Assuming you buy the stock in one trade, give a condition for buying the stock.
2. On average how many minutes will pass before that condition holds (expression or approximation is fine)?
3. If you could split up the trade, i.e, buy different amounts at different minutes, what would you do differently?

13 comments:

As best price is not defined, I assume that the the strategy should achieve the minimum expected value of the stock over all possible strategies.

1) suppose instead of 100 minutes, there was only 1 minute. Then u cant do anything. U will take what the price is (coz u have to buy the stock). Let Sn denote the random variable whose expected value gives us the expected value of the strategy should the time it starts from is the nth minute.

Let the value of stock at the ith minute be Xi. (all are iid uniform 1 to 100)

Hence S100=X100 => E[S100]= 50.5Now suppose we want to calculate S99. We should buy it at the 99th time interval if and only if X99<=E[S100] coz if X99 > E[S100] then we can always skip the transaction and still get a better expected price at t=100HenceS99=min(X99,E[S100])Similarly we can continue the argument to get..S98 = min(X98,E[S99])...S1 = min (X1,E[S2]) and hence the best expected value of the stock that we can buy is E[S1]

3) Here we assume that at the end of the 100th minute we want exactly a fixed amount of the stock.(which was pre-decided). i,e we do not buy a lot of shares exceeding the no of shares we decided, if the price of the shares becomes very less. If we were able to do this, its easy to see that we could get a value as close as possible to min(X1,X2,...,X100)

clearly S100 is the only possible strategy for the 100th minute. Now suppose we get X99. Now if X99E[S100] then if we split, we are still making a loss coz we should buy the stock at the 100th minute.

If X99=E[S100] it does not matter whether we split or not, so we would want to get the stock as early as possible, so we buy at 99th minute. Hence the strategy at 99th minute is the same as in part 1 even if we are allowed to split. Continuing this logic we see that it does not make sense to split at any time. We should just follow the strategy in 1)

I solved this using excel the results i present in form of a list of tuples i.e. if Ith tuple is (k,m) look for prices in range[1,k] for m turns as soon as you get the price u r looking for u stop.{(2,25),(3,23),(4,13),(5,7),(6,6),(7,4),(8,3),(9,2),(10,2),(11,2),12,13,14,15,16,18,20,22,26,30,38,50,100)

For part 3, the "quant" solution would be:Note this is essentially an American option, it’s suboptimal if one does not exercise, or partially exercise the option when the optimal stopping condition is met. hence the freedom of splitting up the trade gives no difference.

pratik's strategy has property that if it gives a price A and any other strategy gives price B, then E[A] <= E[B]. Can we also say that most of the time A <= B i.e. probability [A<=B] >= probability [B<=A].

Instead of achieving the minimum expected value one should have a strategy as follows:if the probability that, in the future, I get a value better than the current value is greater than 0.5 than i should not buy the stock otherwise I should buy the stock.

So I should choose a threshold such that this probability is exactly = 0.5.

So the threshold for each time t should be100 / (2^(1/(100-t)))

I think the expected value strategy should only be followed when you can play the game many times over and then the expected value will be +ve.

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I am an early stage technology investor at Nexus Venture Partners. Prior to this, I was a 3x product entrepreneur. Prior to this, I worked as a private equity analyst at Blackstone and as a quant analyst at Morgan Stanley. I graduated from Department of Computer Science and Engineering of IIT Bombay. I enjoy Economics, Dramatics, Mathematics, Computer Science and Business.

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